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Natural Logarithms
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828.
Learning Objective

Recall the basic properties and uses of the natural logarithm
Key Points

The natural logarithm is the logarithm with base equal to e.

Also known as Euler's number, e is an irrational number that often appears in natural relationships in pure math and science.

The number e and the natural logarithm have many applications in calculus, number theory, differential equations, complex numbers, compound interest, and more.
Terms

e
The base of the natural logarithm, 2.718281828459045…

natural logarithm
The logarithm in base e; either the function that given
$x\!$ returns$y\!$ such that$e^y = x\!$ , or the value of$y\!$ .
Full Text
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. The natural logarithm is the logarithm with base equal to e.
The function slowly grows to positive infinity as x increases and rapidly goes to negative infinity as x approaches 0 ("slowly" and "rapidly" as compared to any power law of x); the yaxis is an asymptote, as seen in .
Also known as Euler's number, e is an irrational number representing the limit of:
as n approaches infinity. In other words, e is the sum of 1 plus 1/1 plus 1/(1*2) plus 1/(1*2*3), and so on.
The number e has many applications in calculus, number theory, differential equations, complex numbers, compound interest, and more. It also is extremely useful as a base in logarithms; so useful that the logarithm with base e has its own name (natural logarithm) and symbol. Here is the proper notation for the natural logarithm of x:
The natural logarithm is so named because unlike 10, which is given value by culture and has minimal intrinsic use, e is an extremely interesting number that often "naturally" appears, especially in calculus.
The inverse of the natural log appears, for example, upon differentiating a logarithm of any base:
Outside of calculus, the natural logarithm can be used to relate 1, e, i, and π, four of the most important numbers in mathematics:
Key Term Reference
 Interest
 Appears in this related concepts: Times Interest Earned Ratio, Interest Compounded Continuously, and Tax Considerations
 asymptote
 Appears in this related concepts: Curve Sketching, Infinite Limits, and Introduction: Polynomial and Rational Functions and Models
 base
 Appears in this related concepts: Overview of the AcidBase Properties of Salt, Integer Exponents, and Balancing Redox Equations
 complex
 Appears in this related concepts: The ComplexNumber System, Electron Transport Chain, and Basic Techniques in Protein Analysis
 complex numbers
 Appears in this related concepts: Addition, Subtraction, and Multiplication, Phasors, and Complex Numbers
 compound
 Appears in this related concepts: Substances and Mixtures, Compound Inequalities, and Using Hyphens
 compound interest
 Appears in this related concepts: Calculating Future Value, Calculating Present Value, and MultiPeriod Investment
 constant
 Appears in this related concepts: Inverse Variation, Combined Variation, and Direct Variation
 equation
 Appears in this related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 exponent
 Appears in this related concepts: Simplifying Expressions of the Form log_a a^x and a(log_a x), Logarithms of Powers, and Scientific Notation
 function
 Appears in this related concepts: Inverse Functions, Limit of a Function, and Modal Mixture
 irrational number
 Appears in this related concepts: Zeroes of Polynomial Functions with Real Coefficients, Domain of a Rational Expression, and Rational Coefficients
 logarithm
 Appears in this related concepts: Derivatives of Logarithmic Functions, Converting between Exponential and Logarithmic Equations, and Special Logarithms
 yaxis
 Appears in this related concepts: The Cartesian System, SlopeIntercept Equations, and Quadratic Equations and Quadratic Functions
Sources
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Cite This Source
Source: Boundless. “Natural Logarithms.” Boundless Algebra. Boundless, 03 Jul. 2014. Retrieved 18 Apr. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/exponentsandlogarithms5/graphinglogarithmicfunctions37/naturallogarithms1775881/